reserve x,y,X,Y for set,
  k,l,n for Nat,
  i,i1,i2,i3,j for Integer,
  G for Group,
  a,b,c,d for Element of G,
  A,B,C for Subset of G,
  H,H1,H2, H3 for Subgroup of G,
  h for Element of H,
  F,F1,F2 for FinSequence of the carrier of G,
  I,I1,I2 for FinSequence of INT;

theorem Th18:
  rng F c= carr H implies Product(F) in H
proof
  defpred P[FinSequence of the carrier of G] means rng $1 c= carr H implies
  Product $1 in H;
A1: now
    let F,d;
    assume
A2: P[F];
    thus P[F^<*d*>]
    proof
A3:   rng F c= rng(F ^ <* d *>) & Product(F ^ <* d *>) = Product(F) * d
      by FINSEQ_1:29,FINSOP_1:4;
A4:   rng<* d *> = {d} & d in {d} by FINSEQ_1:39,TARSKI:def 1;
      assume
A5:   rng(F ^ <* d *>) c= carr H;
      rng<* d *> c= rng(F ^ <* d *>) by FINSEQ_1:30;
      then rng<* d *> c= carr H by A5;
      then d in H by A4,STRUCT_0:def 5;
      hence thesis by A2,A5,A3,GROUP_2:50,XBOOLE_1:1;
    end;
  end;
A6: P[<*> the carrier of G]
  proof
    assume rng <*> the carrier of G c= carr H;
    1_G in H by GROUP_2:46;
    hence thesis by Th8;
  end;
  for F holds P[F] from FINSEQ_2:sch 2(A6,A1);
  hence thesis;
end;
